Exact $U\left(1\right)\mathrm{}$ symmetric cosmologies with local mixmaster dynamics

By applying a standard solution generating technique, we transform an arbitrary vacuum Mixmaster solution on $S^3\times \mathbf{R}$ to a new solution which is spatially inhomogeneous. We thereby obtain a family of exact, spatially inhomogeneous, vacuum spacetimes which exhibit Belinskii-Khalatnikov-Lifshitz oscillatory behavior. The solutions are constructed explicitly by performing the transformations on numerically generated, homogeneous Mixmaster solutions. Their behavior is found to be qualitatively like that seen in previous numerical simulations of generic $U\left(1\right)\mathrm{}$ symmetric cosmological spacetimes on $T^3\times \mathbf{R}.$